How does the "Married filing jointly and your spouse is covered by a plan at work" IRA deduction work?

Meaning the sum of all contributions to retirement accounts for us would be $19,500 for the 401(k) and $6,000 each for two Roth IRAs and one traditional IRA in my spouse's name?

Each person under the age of 50 is entitled to contribute up to the lesser of (earned income shown on the person's tax return and $6K) to IRAs for a given year. If a married couple is filing a joint tax return and at least one of them has earned income for that year, the sum total of what can be contributed to IRAs is limited to the lesser of the total earned income shown on the joint return and $12K, with each spouse being limited at most $6K. That is, if the two spouses together have at least $12K of earned income (and are filing a joint tax return), then each spouse can contribute $6K to their respective IRAs. If the total earned income is less than $12K, then the total amount contributable to IRAs by the two spouses is capped at their total earned income, and each spouse's contribution to his/her IRA is limited to $6K as well. Note that the take-home pay might be less than $12K because of tax withholding etc and so the $12K sent to the IRAs might be partly from earned income and partly from non-earned income such as interest, dividends, capital gains etc or from savings, but that is OK; all that matters is that there is $12K in earned income (e.g. wages, commissions on sales, etc), not what the take-home pay is. As a final note, people over 50 are allowed to contribute more than $6K, but that too is capped, and I have ignored this to avoid cluttering up the discussion.

How that $6K or lesser amount is split into Roth versus Traditional IRAs is up to the individual subject to various special rules (e.g. Roth IRA contributions are prohibited to high earners). In short, the OP is not permitted do what the OP says he wants to do; viz., contribute a total of $18K to IRAs in his name and his spouse's name for 2020.

Everyone is entitled to make contributions to a Traditional IRA (subject to having earned income) but high earners cannot deduct their contributions; making contributions is OK, deducting them on your tax return is not allowed. Note that deductability of Traditional IRA contributions depends on the joint income declared on the joint return; for spouses filing separate tax returns (MFS status) the rules are different. Whether it is wise to make nondeductible contributions is a matter of some controversy on this forum. When a Traditional IRA owner takes a distribution from the IRA, no income tax is due on that part of the distribution that is attributable to the nondeductible contributions, if any. And No, you cannot withdraw just the nontaxable part first and leave the taxable part for later; every distribution (including the very last one that empties out the IRA) consists in part of the nontaxable contributions and in part of taxable income (previously deducted contributions plus all earnings inside the IRA, including earnings attributable to the nondeductible contributions). Finally, as far as the IRS is concerned, you have just one IRA regardless of how many different custodians and mutual funds or stocks and bonds it is invested in, and so taking out money from a different account than the one you made nondeductible contributions to doesn't help.